Problem 5 - Algebra A Combined Function

Question

Write each as an exponential equation. $$ \log _{10} 1000=3 $$

Step-by-Step Solution

Answer

The exponential equation is \( 10^3 = 1000 \).

Step 1: Understanding the Logarithm Equation

The equation given is \( \log_{10} 1000 = 3 \). This represents a logarithmic relationship between the base (10), the result of exponential multiplication (1000), and the exponent (3). The goal is to convert this equation into an exponential form.

Step 2: Rewriting the Logarithm Equation

In a logarithm \( \log_b a = c \), it can be rewritten in its equivalent exponential form \( b^c = a \). Here, the base \( b \) is 10, the result \( a \) is 1000, and the exponent \( c \) is 3. Therefore, we rewrite it as \( 10^3 = 1000 \).

Step 3: Confirming the Exponential Form

Finally, let's verify that \( 10^3 = 1000 \) holds true. Calculate \( 10 \times 10 \times 10 = 100 \times 10 = 1000 \). This confirms our exponential equation is correct.

Key Concepts

Logarithms

A logarithm is a powerful mathematical tool that helps us solve exponential equations by expressing the power to which a given number, the base, must be raised to produce a specific number. When you see an equation like \( \log_{10} 1000 = 3 \), it tells you that 10 must be multiplied by itself 3 times to equal 1000.

• The "base" of the logarithm is the number you repeatedly multiply. In the equation \( \log_{10} \), the base is 10.
• The "logarithm" itself (in this case, 3) is the exponent, indicating how many times \( 10 \) is used as a factor.
• The "result" is the outcome of this multiplication; here, it's 1000.

Understanding logarithms lets us reverse-engineer our way to knowing the exponent used in exponential growth or decay without direct calculation. This concept simplifies the process of working with large numbers and solving equations involving powers.

Base and Exponent

The relationship between base and exponent is at the heart of both logarithmic and exponential expressions. In any expression of the form \( b^c = a \), \( b \) is the **base**, referencing the number that is being multiplied.

• Consider \( 10^3 \), where 10 is the base. It signifies that 10 will be used as a factor.

The **exponent**, denoted as \( c \) in the expression \( b^c \), tells us how many times the base is multiplied by itself.

• In our example, the exponent is 3, indicating the multiplication \( 10 \times 10 \times 10 \).

With understanding base and exponent, you can decipher any expression in exponential form. The base gives the repeated quantity, and the exponent tells you how extensive that repetition is.

Exponential Form

Exponential form is a straightforward way to represent numbers as it compactly conveys repeated multiplication. Converting from a logarithmic form to an exponential form helps make sense of large, seemingly complex numbers.

• From \( \log_{10} 1000 = 3 \), we derive the exponential form \( 10^3 = 1000 \).

In the exponential form, the base (10) and the exponent (3) combine to give the product (1000), telling you everything you need about the number's magnitude. Using exponential form is particularly useful in fields like science, where it deals with real-world phenomena involving growth patterns, like population growth or radioactive decay.

• By mastering exponential form, you can easily expand and simplify calculations, and clearly grasp large numerical concepts.
• With a simple conversion method from logarithmic to exponential, you unlock a comprehensive understanding of numerical relationships.